Mean-periodic function

In mathematical analysis, the concept of a mean-periodic function is a generalization introduced in 1935 by Jean Delsarte[1][2] of the concept of a periodic function. Further results were made by Laurent Schwartz and J-P Kahane.[3][4]

Definition

Consider a continuous complex-valued function f of a real variable. The function f is periodic with period a precisely if for all real x, we have f(x) − f(x − a) = 0. This can be written as

${\displaystyle \int f(x-t)\,d\mu (t)=0\qquad \qquad (1)}$

where ${\displaystyle \mu }$ is the difference between the Dirac measures at 0 and a. The function f is mean-periodic if it satisfies the same equation (1), but where ${\displaystyle \mu }$ is some arbitrary nonzero measure with compact (hence bounded) support.

Equation (1) can be interpreted as a convolution, so that a mean-periodic function is a function f for which there exists a compactly supported (signed) Borel measure ${\displaystyle \mu }$ for which ${\displaystyle f*\mu =0}$.[4]

There are several well-known equivalent definitions.[2]

Relation to almost periodic functions

Mean-periodic functions are a separate generalization of periodic functions from the almost periodic functions. For instance, exponential functions are mean-periodic since exp(x+1) − e.exp(x) = 0, but they are not almost periodic as they are unbounded. Still, there is a theorem which states that any uniformly continuous bounded mean-periodic function is almost periodic (in the sense of Bohr). In the other direction, there exist almost periodic functions which are not mean-periodic.[2]

Some basic properties

If f is a mean periodic function, then it is the limit of a certain sequence of exponential polynomials which are finite linear combinations of term t^^n exp(at) where n is any non-negative integer and a is any complex number; also Df is a mean periodic function (ie mean periodic) and if h is an exponential polynomial, then the pointwise product of f and h is mean periodic).

If f and g are mean periodic then f + g and the truncated convolution product of f and g is mean periodic. However, the pointwise product of f and g need not be mean periodic.

If L(D) is a linear differential operator with constant co-efficients, and L(D)f = g, then f is mean periodic if and only if g is mean periodic.

For linear differential difference equations such as Df(t) - af(t - b) = g where a is any complex number and b is a positive real number, then f is mean periodic if and only if g is mean periodic.[5]

Applications

In work related to the Langlands correspondence, the mean-periodicity of certain (functions related to) zeta functions associated to an arithmetic scheme have been suggested to correspond to automorphicity of the related L-function.[6] There is a certain class of mean-periodic functions arising from number theory.

See also

• 　almost periodic functions

References

1. Delsarte, Jean (1935). "Les fonctions moyenne-périodiques". Journal de Mathématiques Pures et Appliquées. 17: 403–453.
2. Kahane, J.-P. (1959). Lectures on Mean Periodic Functions (PDF). Tata Institute of Fundamental Research, Bombay.
3. Malgrange, Bernard (1954). "Fonctions moyenne-périodiques (d'après J.-P. Kahane)" (PDF). Séminaire Bourbaki (97): 425–437.
4. Schwartz, Laurent (1947). "Théorie générale des fonctions moyenne-périodiques" (PDF). Ann. of Math. 48 (2): 857–929. doi:10.2307/1969386. JSTOR 1969386.
5. Laird, P. G. (1972). "Some properties of mean periodic functions". Journal of the Australian Mathematical Society. 14 (4): 424–432. doi:10.1017/s1446788700011058. ISSN 0004-9735.
6. Fesenko, I.; Ricotta, G.; Suzuki, M. (2012). "Mean-periodicity and zeta functions". Annales de l'Institut Fourier. 62 (5): 1819–1887. arXiv:0803.2821. doi:10.5802/aif.2737.